Descriptional Complexity of Three-Nonterminal Scattered Context Grammars: An Improvement

TL;DR

This paper improves the understanding of three-nonterminal scattered context grammars by showing that the maximum number of nonterminals rewritten simultaneously can be bounded by a small constant, independent of language specifics.

Contribution

It demonstrates that the maximal number of nonterminals rewritten in a derivation step can be limited by a small constant, regardless of language properties.

Findings

Maximal number of nonterminals rewritten is bounded by a small constant.

The result is independent of alphabet size and language structure.

Enhances the theoretical understanding of scattered context grammars.

Abstract

Recently, it has been shown that every recursively enumerable language can be generated by a scattered context grammar with no more than three nonterminals. However, in that construction, the maximal number of nonterminals simultaneously rewritten during a derivation step depends on many factors, such as the cardinality of the alphabet of the generated language and the structure of the generated language itself. This paper improves the result by showing that the maximal number of nonterminals simultaneously rewritten during any derivation step can be limited by a small constant regardless of other factors.